Thermodynamics done right: (lack of) pedagogical material

 

[This post is a work in progress and will be updated with time, please feel free to add your favourite reference in the comments and I will discuss it]

From time to time I like to say in my talks, and write in my papers, that we now have a modern theory of nonequilibrium thermodynamics based on Markov processes that covers most of the “old-style” thermodynamic phenomena and that greatly encompasses our old understanding of nonequilibrium phenomena, and that time is ripe for it to supplant the way we teach thermodynamics in physics courses, which is in a quite shameful state. While the physical insights that made this possible are already quite old (dating back e.g. to the Einstein relation), only recently a vast community reasoning in these terms has coalesced. However, unfortunately, so far vary little effort within this community has been made to create pedagogical material. I personally fantasize of doing my bit, maybe writing some lecture notes, or even a monograph. However, this humongous task looks prohibitive, given that I’m at the same time a procrastinator and a perfectionist.  For the moment I can only list what to me seem to be the best materials to getting acquainted with “modern” thermodynamics.

The only broad review on the topic is

U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Progr. Phys. 75, 12600112 (2012) [arXiv:1205.4176]

However, I do not suggest it for learning the theory. While doing a good job in surveying trends, topics, and methods, it was not written with a pedagogical intent, so it turns out being a loose collection. This I would only suggest for a later reading when one already masters a few of the technical tools, and wants to get to the frontier of research today.

When engaging with thermodynamics based on stochastic processes, it will be convenient to choose one particular formalism, either discrete (based on the Master Equation) or continuous  (based on diffusion equations). Results are interchangeable but it’s good to focus on one only at the start. On the discrete side, which is my favourite, this short review by Van den Broeck and Esposito, based on lecture notes of a course, is simple and accessible and gives a bit of an overview:

C. Van den Broeck and M. Esposito, Ensemble and trajectory thermodynamics: A brief introduction, Physica A 418, 6 (2015) [arXiv:1403.1777].

My own Ph. D. thesis might be a slightly more technical guidance for the analysis, in particular Chapters 3 and 4. What it’s nice about them is that I try to separate what is the dynamics of Markov processes, what is the thermodynamics associated to them, and furthermore (in Chapter 1-2) which aspects of nonequilibrium thermodynamics are only due to the geometry/topology of the state space with no particular reference to the fact that there is an underlying Markov process. I also try to avoid the plethora of thermodynamic potentials that haunt thermodynamic talking. It can be found here (I hold no responsibility for what I was writing 6 7 years ago!):

M. Polettini, Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics [link]

In general, Ph. D. thesis might be a good place to look at for introductions on the topic. Chapter 8 of Takahiro Sagawa’s published thesis does a good job in reviewing the fluctuation theorems, and the previous chapters can be used to get an overview of so-called Maxwell-demons revisited:

T. Sagawa, Thermodynamics of Information Processing in Small Systems (Springer, 2012) [link]

On the diffusion side, one could make a blend of Van Kampen’s and Ken Sekimoto’s books, the first describing the dynamics of diffusion equations, the second describing the thermodynamics.

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland)
K. Sekimoto, Stochastic Energetics (Berlin: Springer)

For more recent research trends, the New Journal of Physics hosted a special issue, presented by the editors here:

C. Van den Broeck, S.-i. Sasa, and U. Seifert, Focus on stochastic thermodynamics, New J. Phys. 18, 020401 (2016) [link]

However, all of these references somehow fall short in giving an explicit systematic connection to classic topics in thermodynamics and Equilibrium Statistical Mechanics, such as the ideal gas law, the Carnot cycle, etc. This is mainly due because these references look at the future of thermodynamics at the nanoscale, rather than covering the past of thermodynamics, but it’s clear that if any serious claim has to be made about this being the way thermodynamics is taught, we seriously need to tackle the problem of incorporating the most important achievements of thermodynamics and Equilibrium Statistical Mechanics in an elegant and self-consistent fashion. This is in principle possible, but it has to be made explicitly. So, for the moment, the connection to thermodynamics has to be traced back piece by piece by the engaged student.

[To be added soon: refs. on Large Deviation Theory, interacting particle models]

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